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SUMMARYThe Neumann-Neumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finiteelement discretizations of difficult problems with strongly discontinuous coefficients (De Roeck and Le Tallec, 1991). However, this algorithm suffers from the need to solve in each iteration an inconsistent singular problem for every subdomain, and its convergence deteriorates with increasing number of subdomains due to the lack of a coarse problem to propagate the error globally. We show that the equilibrium conditions for the singular problems on subdomains lead to a simple and natural construction of a coarse problem. The construction is purely algebraic and applies also to systems such as those that arise in elasticity. A convergence bound independent of the number of subdomains is proved and results of computational tests are reported.

Abstract. We describe the physical model, numerical algorithms, and software structure of a model consisting of the Weather Research and Forecasting (WRF) model, coupled with the fire-spread model (SFIRE) module. In every time step, the fire model inputs the surface wind, which drives the fire, and outputs the heat flux from the fire into the atmosphere, which in turn influences the atmosphere. SFIRE is implemented by the level set method, which allows a submesh representation of the burning region and a flexible implementation of various kinds of ignition. The coupled model is capable of running on a cluster faster than real time even with fine resolution in dekameters. It is available as a part of the Open Wildland Fire Modeling (OpenWFM) environment at http://openwfm.org, which contains also utilities for visualization, diagnostics, and data processing, including an extended version of the WRF Preprocessing System (WPS). The SFIRE code with a subset of the features is distributed with WRF 3.3 as WRF-Fire.

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Dedicated to Professor Ivo Marek on the occasion of his 70th birthday. AbstractA convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The preconditioner is formulated as an Additive Schwarz method and analyzed by building on existing results for Balancing Domain Decomposition. The main result is a bound on the condition number based on inequalities involving the matrices of the preconditioner. Estimates of the usual form C(1 + log 2 (H/h)) are obtained under the standard assumptions of substructuring theory. Computational results demonstrating the performance of method are included.

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